Gordon Laing wrote:
To avoid the aperture changing, shoot in Creative Manual mode, choose Manual or Aperture Priority and select an f-number of f5.8 or bigger.
Unfortunately, that won't completely eliminate the issue. The "problem" is that this is primarily a stills (photography) camera and lens (read: not designed to zoom while taking a photograph), not a motion picture camera and lens (which usually are designed to zoom during a shot), which only costs $1,500. As compared to, say, an AF100 ($5,000) and Red Pro 18-85mm ($10,000) which together cost ten times as much! Nevermind a Zeiss Master Zoom 16-110mm which, by itself, is $60,000!
If you really need to zoom during a shot, you're better off using a camera and lens designed specifically for video (like the HMC40). Alternatively, the 14-45mm (note: NOT the 14-42mm) and 45-200mm lenses have less of an issue because, individually, they cover a shorter focal length.